代写MODULE 14.2、代做CS/Python程序、代做Random Walk、C/C++设计代做

日期：2018-11-02 10:42

MODULE 14.2

Solidification—Let’s Make It Crystal Clear!

Prerequisite: Module 9.5, “Random Walk.”

Introduction

What do snowflakes and steel have in common? At first glance, we probably would

say, not much. However, if we could look closely enough, we would see that they

both are crystalline, possessing amazing structural similarities. Each is made of treelike

structures called dendrites, which are formed as substance cools during the

process of solidification.

Snowflakes are composed of one or more snow crystals. Each crystal is built of

water molecules arranged in a very specific, hexagonal lattice. These crystals form

in the clouds by the condensation of water vapor into ice. At first, while very small,

the crystals form as hexagonally shaped prisms, following the original, molecular

symmetry. The edges of the facets of this prism grow out more rapidly than the facets

themselves, leading to the formation of “limbs.” These limbs may, and usually

do, produce other branches, leading to the dendrite, or treelike, forms.

A number of factors determine the precise shape of the crystal, but temperature is

the primary influence. As snowflakes blow and fall through the clouds, they encounter

significant variations in temperature, humidity, and pressure. Each snowflake

tends to have different environmental “experiences,” which lead to the development

of different shapes. Why snowflake shape is so temperature dependent is not completely

understood (Libbrecht).

The solidification of snowflakes is fascinating, but the process of solidification

has an impressive array of manufacturing applications. Despite the increased use of

plastics, think of all the things we use everyday that are metal. Used to produce everything

from soda cans to car engines, these metals and alloys are formed from

liquids that have “frozen,” or solidified. Solidification, therefore, is an important

process for generating metal products as well as snowflakes.

Dendrites form within the molten metals/alloys as they solidify during the casting

process. These dendrites vary greatly in shape, size, and orientation. Furthermore,

In order to view this proof accurately, the Overprint Preview Option must be checked

in Acrobat Professional or Adobe Reader. Please contact your Customer Service Representative

if you have questions about finding the option.

Job Name: Cyan = PMS 300/356585t

700 Module 14.2

the individual dendrites interconnect in various ways to generate a series of intricate

microstructures. These individual and collective variations greatly influence the

structural qualities (e.g., strength and flexibility) of the product (Glicksman et al.

1991). There are numerous horror stories of castings that have broken apart from

internal defects that originated from thermal stresses occurring during solidification

(Seetharamu et al. 2001). According to scientists, we would be able to understand

(and, therefore, control) the properties of materials that solidify dendritically better

if we could develop effective computational models of the behavior of individual

dendrites.

Under the influence of Earth’s gravity, liquid metal is subject to the influence of

convective currents as it cools. These currents significantly alter the growth of the

dendrites, which makes modeling of “normal” dendritic growth and the effects of

convective currents on such growth virtually impossible. Confronting this difficulty,

the National Aeronautics and Space Administration (Glicksman et al. 1991) has

teamed with scientists at Rensselaer Polytechnic Institute in the Isothermal Dendritic

Growth Experiment (IDGE). Experiments in this program, conducted in conditions

of low gravity that Earth orbit offers, have already shed tremendous light on dendritic

growth. For instance, scientists, using IDGE data, will be able to separate the

effects of convection from other factors that impact solidification of metals and alloys.

Such information will go far to improve computational models, which should

guide us to improved industrial production of various metals/alloys.

Projects

1. a. We can use the technique of diffusion-limited aggregation (DLA) to

build a dendritic structure. In one form of the algorithm, a seed, or initial

location for the developing dendritic structure, is in the middle of an

m × m launching rectangle. This launching rectangle is a region in the

middle of an n × n grid, where m < n. For example, m might be 16 and n

might be 40. One at a time, “particles” are released from random positions

on the launching rectangle boundary to go on random walks. If the walker

comes in contact with another particle (i.e., a neighbor to its north, east,

south, or west), with a designated sticking probability, the walker adheres

to the particle, resulting in a larger structure. If the walker travels too

close to the boundary of the grid, the simulation deletes that walker and

releases another random walker from the launching rectangle. Use the

DLA algorithm to develop a simulation to generate dendritic structures,

with the number of particles for the structure as a parameter (Panoff

2004).

b. Develop a visualization that shows the simulation one step at a time, including

the random walks. Develop another animation that shows only the

particles as a new particle attaches to the growing structure. An attractive

enhancement is for the color of the particle to be a function of its distance

from the seed. (Follow the link “Simple DLA Example” at the Shodor

website for an example of such a simulation with animation (The Shodor

Educational Foundation 2002).)

In order to view this proof accurately, the Overprint Preview Option must be checked

in Acrobat Professional or Adobe Reader. Please contact your Customer Service Representative

if you have questions about finding the option.

Job Name: Cyan = PMS 300/356585t

Additional Cellular Automata, Agent-Based and Matrix Projects 701

c. Run the simulation and visualization a number of times for several different

sticking probabilities. Discuss the impact of the sticking probabilities

on the resulting structures.

2. Repeat Project 1 considering the eight surrounding cells as a walker’s nearest

neighbors.

3. Repeat Project 1, Parts a and b, where the sticking probability is 0.33 for

contact with one particle, 0.67 for simultaneous contact with two particles,

and 1.0 for contact with three. Run the simulation a number of times and

discuss the results (Panoff 2004).

4. a. Repeat Project 2, Parts a and b, where the sticking probability is based on

the number of particles the walker contacts simultaneously. Run the simulation

a number of times and discuss the results (Panoff 2004).

b. Adjust the situation so that the sticking probability is 0.1 for contact with

one particle, 0.5 for two particles, and 0.9 for three or more particles. Run

the simulation and animation a number of times and discuss the results

(Panoff 2004).

c. Adjust the situation so that the sticking probability is 0.01 for contact with

one or two particles, 0.03 for three particles, and 1.0 for more than three

particles. Run the simulation a number of times and discuss the results

(Panoff 2004).

5. Repeat Project 1, Parts a and b, where the sticking probability is greater for

bonds continuing in a straight line. For example, a walker is more likely to

adhere to a north neighbor if that particle is stuck to a particle to its north.

Similar situations exist for the other directions. Run the simulation a number

of times and discuss the results (Panoff 2004).

6. Repeat Project 5, considering the eight surrounding cells as a walker’s nearest

neighbors (Panoff 2004).

7. Changing conditions affect crystalline formation and cause a great variety in

the shapes. During a simulation, we can vary the sticking probability to indicate

such changing conditions. Do Project 2, starting with sticking probabilities

as in Project 4, Part b. After forming an aggregate with a specified number

(such as 100) of particles, use sticking probabilities, as in Project 4c, for

a specified number (such as 100) of particles; then change to a different sticking

probability configuration (Panoff 2004).

8. Repeat any of Projects 1–6, considering the impact of wind or gravity on

dendritic growth by having the walker travel with a greater probability in a

particular direction (Shodor 2002).

9. Repeat any of Projects 1-8, using a launching circle of radius m instead of a

launching rectangle. (Follow the link “Diffusion Limited Aggregation Calculator”

at the Shodor website for such a simulation example (The Shodor

Educational Foundation 2002).)

10. Repeat any of Projects 1–8, using a launching circle instead of a launching

rectangle, of radius 2rmax, where rmax is the radius of the structure so far. Delete

a walker if it travels too close to the boundary of the grid or beyond a

distance of 3rmax from the seed. Such adjustments should speed the simulation

(Gould and Tobochnik 1988).

11. Do Project 10, with the following additional adjustment to speed the simulation

by having larger step sizes further away from the structure: If a walker is

In order to view this proof accurately, the Overprint Preview Option must be checked

in Acrobat Professional or Adobe Reader. Please contact your Customer Service Representative

if you have questions about finding the option.

Job Name: Cyan = PMS 300/356585t

702 Module 14.2

at a distance r > rmax + 4 from the seed, where rmax is the radius of the structure

so far, then have step sizes of length r – rmax – 2; otherwise, have step

sizes of length 1 (Gould and Tobochnik 1988).

12. Repeat any of Projects 1 or 2, considering accumulation on a structure, such

as the deposit of snow on a tree. Have the seed be a triangular tree-like structure

or other type of structure on the bottom of the grid. Release random

walkers from the north end of the grid with a greater likelihood of traveling

south (Panoff 2004).

References

Glicksman, Martin E., M. B. Koss, R. C. Hahn, Ana Cris R. Veloso, A. Rojas, and E.

Winsa. 1991. “Scientific Basis for the Isothermal Dendritic Growth Experiment:

A USMP-2 Space Flight Experiment.” In Materials Science Forum, 77: 51–60.

Gould, Harvey, and Jan Tobochnik. 1988. An Introduction to Computer Simulation

Methods, Applications to Physical Systems, Part 2. Reading, MA: Addison-Wesley:

695.

Libbrecht, Kenneth G. “Snowflake Primer—The Basic Facts About Snowflakes and

Snow Crystals.” California Institute of Technology. http://www.its.caltech

.edu/~atomic/snowcrystals/primer/primer.htm (accessed January 1, 2013)

Panoff, Robert. 2004. “Diffusion Limited Aggregation.” Educational Materials for

Undergraduate Compuational Science. Capital University. http://www.capital

.edu/cs-computational-science/ (accessed January 1, 2013)

Seetharamu, K. N., R. Paragasam, Ghulam A. Quadir, Z. A. Zainal, P. Sthaya Prasad,

and T. Sundararajan. 2001. “Finite Element Modeling of Solidification Phenomena.”

Sadhana, 26 (Parts 1 and 2): 103–120.

The Shodor Educational Foundation. 2002. “Software—Diffusion Limited Aggregation

Calculator.” Computational Science Education Reference Desk. http://www

.shodor.org/refdesk/Resources/Models/DLA/ (accessed January 1, 2013)